Abstract: Given a weighted graph $G=(V, E)$ with weight $w: E \rightarrow \mathbb{Z}^{+}$, a $k$-cycle transversal is an edge subset $A$ of $E$ such that $G-A$ has no $k$-cycle. The minimum weight of $k$ cycle transversal is the weighted transversal number on $k$-cycle, denoted by $\tau_{k}\left(G_{w}\right)$. In this paper, we design a $(k-1 / 2)$-approximation algorithm for the weighted transversal number on $k$-cycle when $k$ is odd. Given a weighted graph $G=(V, E)$ with weight $w: E \rightarrow \mathbb{Z}^{+}$, a $k$-clique transversal is an edge subset $A$ of $E$ such that $G-A$ has no $k$-clique. The minimum weight of $k$-clique transversal is the weighted transversal number on $k$-clique, denoted by $\widetilde{\tau_{k}}\left(G_{w}\right)$. In this paper, we design a $\left(k^{2}-k-1\right) / 2-$ approximation algorithm for the weighted transversal number on $k$-clique. Last, we discuss the relationship between $k$-clique covering and $k$-clique packing in complete graph $K_{n}$.

Key words: k-cycle transversal, k-clique transversal, Approximation algorithms